Probability
Bayes Theorem
Grade 12
Question:
<p>A hat contains a number of cards with 30% white on both sides, 50% black on one side and white on the other side, 20% black on both sides. The cards are mixed up, and a single card is drawn at random and placed on the table. Its upper side shows up black. The probability that its other side is also black is</p>
<p>(1) 2/9</p>
<p>(2) 4/9</p>
<p>(3) 2/3</p>
<p>(4) 2/7</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem: given that the upper side is black, find the probability the card is 'black-black' by considering which cards can show black on top and their frequencies.
<p><strong>Step 1:</strong> Identify the three card types and their probabilities:</p><ul><li>WW (white both sides): 30%</li><li>BW (black one side, white other): 50%</li><li>BB (black both sides): 20%</li></ul><p><strong>Step 2:</strong> Find P(showing black on top). Only BB and BW cards can show black:</p><ul><li>BB card shows black: P = 0.20 × 1 = 0.20</li><li>BW card shows black: P = 0.50 × 1/2 = 0.25</li><li>P(black showing) = 0.20 + 0.25 = 0.45</li></ul><p><strong>Step 3:</strong> Apply Bayes' theorem. We need P(BB | black showing):</p><p>P(BB | black showing) = P(black showing | BB) × P(BB) / P(black showing)</p><p>= (1 × 0.20) / 0.45 = 0.20 / 0.45 = 20/45 = <strong>4/9</strong></p><p><strong>Note:</strong> If the answer key states 2, there may be a transcription issue. The correct probability is <strong>4/9 ≈ 0.444</strong></p>
Correct Answer: 2